The most difficult question, that I have found, is to describe how Shakespeare influenced the world around him. Now I know the obvious, such as entertainment purposes, but are there others I could think to mention?

The better question is how did he not influence it. He added many words and commonly used expressions to the English language: ex., "It's Greek to me"; "salad days"; "budge an inch"; "knitted brows"; and many more. Enotes has several articles about his influence in many areas. Go to http://www.enotes.com/shakespeare-influence.

The two links given above will take you to the eNotes section about Shakespeare's world and influence. If you click on these, you'll find essays about all sorts of areas that might be of help for your report, such as Shakespeare and Warfare, The Sense of History in Renaissance England, Shakespeare and the Humanities, and Shakespeare in Music.

One thing that is definitely worth looking into with regard to how Shakespeare influenced in the world is through his women. His female characters were among the first in literature to be strong-willed, independent-minded, and able to stand on their own two feet. Katherine in The Taming of the Shrew, Viola in Twelfth Night, Beatrice in Much Ado About Nothing are only a few examples.

I agree that Shakespeare presents strong women, but all of the above examples, and more, have destinies tied to men: ie, marriage. Beatrice and Katherine are tamed, Viola marries, Gertrude dies, as does Juliet, on and on. Mothers (as discussed in another post) tend to be cold, distant, or absent.

I am not convinced that Shakespeare was advocating new roles as much as he was exploring the ramifications of a changing society.

I agree that Shakespeare presents strong women, but all of the
above examples, and more, have destinies tied to men: ie,
marriage. Beatrice and Katherine are tamed, Viola marries,
Gertrude dies, as does Juliet, on and on. Mothers (as
discussed in another post) tend to be cold, distant, or
absent.

I am not convinced that Shakespeare was advocating new roles as
much as he was exploring the ramifications of a changing
society.

I have to agree. Women almost seem to be an afterthought or
there because they have to be. Emma Thompson's portrayal of
Beatrice may have been powerful, but is the character truly so?
Think of how poor Hero is treated in the same play. The only really
powerful woman I can think of is Lady Macbeth--and she goes
mad!

I see what you mean, however, I guess I'm looking at it in a different way. Katherine chooses to marry...she's not forced into it, and I don't see her as "tamed". Her speech at the end can be read as a huge joke on all who hear it. Through mutual respect and love, she and Petruchio have come to an understanding.

Viola, though she marries, does so only after living the life of a boy in boy's clothing. She is a nod from Shakespeare what it must be like to be female and be completely under the male thumbs of the time period--father, brother, husband.

Most obviously Shakespeare influenced the world around him. Like you said, one of the biggest ways was through entertainment and bringing pleasure to those he wrote for. I also believe that he helped shape society in a different way. Perhaps not while he was alive, but definitely today he has looked up to as an idol and hero for many playwrights and writers in general. I also believe that, in many of his works, he provides his opinion on different aspects of society and how he thinks it either works then or how it should work in the future. This can be analyzed by most anyone and influence them to possibly make changes about the way they live.